Hi martin greter,

Sorry for late reply to this,

1. I assume that the system should inform group members when an new event is published. Am I right in this assumption?

Try follow my instruction below :
STEP 1 : Update with your Easysocial version to 1.3.23
STEP 2 : Go to your backend > Easysocial > Event > category > select one of the event > add "member transfer type" custom field in your event form (check my first screenshot below) > Save
STEP 3 : Go to your frontend > create an event in the group > select just you added that event custom field in the event category > set "member transfer type" - INVITE > fill in all the event info > save
STEP 4 : All the group member will get the invitation email notification

2. I have tried to invite group members via 'Invite Friends'... Why is this not possible?

Yes, I've tried in my locally, it should work fine. Check my second screenshot.